1  /* mpfr_asin -- arc-sinus of a floating-point number
       2  
       3  Copyright 2001-2023 Free Software Foundation, Inc.
       4  Contributed by the AriC and Caramba projects, INRIA.
       5  
       6  This file is part of the GNU MPFR Library.
       7  
       8  The GNU MPFR Library is free software; you can redistribute it and/or modify
       9  it under the terms of the GNU Lesser General Public License as published by
      10  the Free Software Foundation; either version 3 of the License, or (at your
      11  option) any later version.
      12  
      13  The GNU MPFR Library is distributed in the hope that it will be useful, but
      14  WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
      15  or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU Lesser General Public
      16  License for more details.
      17  
      18  You should have received a copy of the GNU Lesser General Public License
      19  along with the GNU MPFR Library; see the file COPYING.LESSER.  If not, see
      20  https://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
      21  51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */
      22  
      23  #include "mpfr-impl.h"
      24  
      25  int
      26  mpfr_asin (mpfr_ptr asin, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
      27  {
      28    mpfr_t xp;
      29    int compared, inexact;
      30    mpfr_prec_t prec;
      31    mpfr_exp_t xp_exp;
      32    MPFR_SAVE_EXPO_DECL (expo);
      33    MPFR_ZIV_DECL (loop);
      34  
      35    MPFR_LOG_FUNC (
      36      ("x[%Pd]=%.*Rg rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, rnd_mode),
      37      ("asin[%Pd]=%.*Rg inexact=%d", mpfr_get_prec (asin), mpfr_log_prec, asin,
      38       inexact));
      39  
      40    /* Special cases */
      41    if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
      42      {
      43        if (MPFR_IS_NAN (x) || MPFR_IS_INF (x))
      44          {
      45            MPFR_SET_NAN (asin);
      46            MPFR_RET_NAN;
      47          }
      48        else /* x = 0 */
      49          {
      50            MPFR_ASSERTD (MPFR_IS_ZERO (x));
      51            MPFR_SET_ZERO (asin);
      52            MPFR_SET_SAME_SIGN (asin, x);
      53            MPFR_RET (0); /* exact result */
      54          }
      55      }
      56  
      57    /* asin(x) = x + x^3/6 + ... so the error is < 2^(3*EXP(x)-2) */
      58    MPFR_FAST_COMPUTE_IF_SMALL_INPUT (asin, x, -2 * MPFR_GET_EXP (x), 2, 1,
      59                                      rnd_mode, {});
      60  
      61    /* Set x_p=|x| (x is a normal number) */
      62    mpfr_init2 (xp, MPFR_PREC (x));
      63    inexact = mpfr_abs (xp, x, MPFR_RNDN);
      64    MPFR_ASSERTD (inexact == 0);
      65  
      66    compared = mpfr_cmp_ui (xp, 1);
      67  
      68    MPFR_SAVE_EXPO_MARK (expo);
      69  
      70    if (MPFR_UNLIKELY (compared >= 0))
      71      {
      72        mpfr_clear (xp);
      73        if (compared > 0)                  /* asin(x) = NaN for |x| > 1 */
      74          {
      75            MPFR_SAVE_EXPO_FREE (expo);
      76            MPFR_SET_NAN (asin);
      77            MPFR_RET_NAN;
      78          }
      79        else                              /* x = 1 or x = -1 */
      80          {
      81            if (MPFR_IS_POS (x)) /* asin(+1) = Pi/2 */
      82              inexact = mpfr_const_pi (asin, rnd_mode);
      83            else /* asin(-1) = -Pi/2 */
      84              {
      85                inexact = -mpfr_const_pi (asin, MPFR_INVERT_RND(rnd_mode));
      86                MPFR_CHANGE_SIGN (asin);
      87              }
      88            mpfr_div_2ui (asin, asin, 1, rnd_mode);
      89          }
      90      }
      91    else
      92      {
      93        /* Compute exponent of 1 - ABS(x) */
      94        mpfr_ui_sub (xp, 1, xp, MPFR_RNDD);
      95        MPFR_ASSERTD (MPFR_GET_EXP (xp) <= 0);
      96        MPFR_ASSERTD (MPFR_GET_EXP (x) <= 0);
      97        xp_exp = 2 - MPFR_GET_EXP (xp);
      98  
      99        /* Set up initial prec */
     100        prec = MPFR_PREC (asin) + 10 + xp_exp;
     101  
     102        /* use asin(x) = atan(x/sqrt(1-x^2)) */
     103        MPFR_ZIV_INIT (loop, prec);
     104        for (;;)
     105          {
     106            mpfr_set_prec (xp, prec);
     107            mpfr_sqr (xp, x, MPFR_RNDN);
     108            mpfr_ui_sub (xp, 1, xp, MPFR_RNDN);
     109            mpfr_sqrt (xp, xp, MPFR_RNDN);
     110            mpfr_div (xp, x, xp, MPFR_RNDN);
     111            mpfr_atan (xp, xp, MPFR_RNDN);
     112            if (MPFR_LIKELY (MPFR_CAN_ROUND (xp, prec - xp_exp,
     113                                             MPFR_PREC (asin), rnd_mode)))
     114              break;
     115            MPFR_ZIV_NEXT (loop, prec);
     116          }
     117        MPFR_ZIV_FREE (loop);
     118        inexact = mpfr_set (asin, xp, rnd_mode);
     119  
     120        mpfr_clear (xp);
     121      }
     122  
     123    MPFR_SAVE_EXPO_FREE (expo);
     124    return mpfr_check_range (asin, inexact, rnd_mode);
     125  }